For $f(x)=2x+1$ And $g(x)=x^2-7$, Find $(f \cdot G)(x$\].A. $2x^3-x^2-7x-7$ B. $2x^3+x^2-14x-7$ C. $2x^2-13$ D. $3x^2+7x-6$

Introduction

In algebra, multiplying two polynomial functions involves multiplying each term of one function by each term of the other function and then combining like terms. In this article, we will explore how to multiply two polynomial functions, using the given functions $f(x)=2x+1$ and $g(x)=x^2-7$ as examples.

Understanding Polynomial Functions

Before we dive into multiplying the functions, let's briefly review what polynomial functions are. A polynomial function is a function that can be written in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $a_n \neq 0$ and $n$ is a non-negative integer. The terms of a polynomial function are the individual terms that make up the function, such as $2x$ and $1$ in the function $f(x)=2x+1$.

Multiplying the Functions

To multiply the functions $f(x)=2x+1$ and $g(x)=x^2-7$, we need to multiply each term of one function by each term of the other function. This means that we will multiply each term of $f(x)$ by each term of $g(x)$ and then combine like terms.

Step 1: Multiply Each Term of $f(x)$ by Each Term of $g(x)$

To multiply each term of $f(x)$ by each term of $g(x)$, we will use the distributive property of multiplication over addition. This means that we will multiply each term of $f(x)$ by each term of $g(x)$ and then combine like terms.

For example, to multiply the term $2x$ in $f(x)$ by the term $x^2$ in $g(x)$, we will multiply $2x$ by $x^2$ to get $2x^3$. To multiply the term $2x$ in $f(x)$ by the term $-7$ in $g(x)$, we will multiply $2x$ by $-7$ to get $-14x$.

Step 2: Combine Like Terms

After multiplying each term of $f(x)$ by each term of $g(x)$, we will combine like terms. This means that we will add or subtract terms that have the same variable and exponent.

For example, to combine the terms $2x^3$ and $-2x^3$, we will add them together to get $4x^3$. To combine the terms $-14x$ and $7x$, we will subtract them to get $-21x$.

Step 3: Write the Final Answer

After combining like terms, we will write the final answer. This means that we will write the resulting polynomial function in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

For example, if we multiply the functions $f(x)=2x+1$ and $g(x)=x^2-7$, we will get the resulting polynomial function $(f \cdot g)(x) = 2x^3 - 14x - 7$.

Conclusion

In conclusion, multiplying two polynomial functions involves multiplying each term of one function by each term of the other function and then combining like terms. By following the steps outlined in this article, we can multiply two polynomial functions and write the resulting polynomial function in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

Example Solution

Let's use the functions $f(x)=2x+1$ and $g(x)=x^2-7$ as examples to illustrate how to multiply two polynomial functions.

Step 1: Multiply Each Term of $f(x)$ by Each Term of $g(x)$

To multiply each term of $f(x)$ by each term of $g(x)$, we will use the distributive property of multiplication over addition.

$f(x) \cdot g(x) = (2x+1) \cdot (x^2-7)$

$= 2x \cdot x^2 - 2x \cdot 7 + 1 \cdot x^2 - 1 \cdot 7$

$= 2x^3 - 14x + x^2 - 7$

Step 2: Combine Like Terms

After multiplying each term of $f(x)$ by each term of $g(x)$, we will combine like terms.

$2x^3 - 14x + x^2 - 7$

$= 2x^3 + x^2 - 14x - 7$

Step 3: Write the Final Answer

After combining like terms, we will write the final answer.

$(f \cdot g)(x) = 2x^3 + x^2 - 14x - 7$

Answer

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Introduction

In our previous article, we explored how to multiply two polynomial functions using the given functions $f(x)=2x+1$ and $g(x)=x^2-7$ as examples. In this article, we will answer some common questions that students often have when multiplying two polynomial functions.

Q&A

Q: What is the difference between multiplying two polynomial functions and adding or subtracting two polynomial functions?

A: When multiplying two polynomial functions, we multiply each term of one function by each term of the other function and then combine like terms. When adding or subtracting two polynomial functions, we add or subtract the corresponding terms of the two functions.

Q: How do I know which terms to combine when multiplying two polynomial functions?

A: When multiplying two polynomial functions, we combine like terms by adding or subtracting the corresponding terms of the two functions. For example, if we have the terms $2x^3$ and $-2x^3$, we will add them together to get $4x^3$.

Q: Can I multiply two polynomial functions with different variables?

A: Yes, you can multiply two polynomial functions with different variables. For example, if we have the functions $f(x)=2x+1$ and $g(y)=y^2-7$, we can multiply them together to get $(f \cdot g)(x,y) = 2xy^2 - 14y + x^2 - 7$.

Q: How do I know which terms to multiply when multiplying two polynomial functions?

A: When multiplying two polynomial functions, we multiply each term of one function by each term of the other function. For example, if we have the functions $f(x)=2x+1$ and $g(x)=x^2-7$, we will multiply each term of $f(x)$ by each term of $g(x)$ to get $2x \cdot x^2 - 2x \cdot 7 + 1 \cdot x^2 - 1 \cdot 7$.

Q: Can I multiply two polynomial functions with different degrees?

A: Yes, you can multiply two polynomial functions with different degrees. For example, if we have the functions $f(x)=2x+1$ and $g(x)=x^2-7$, we can multiply them together to get $(f \cdot g)(x) = 2x^3 - 14x + x^2 - 7$.

Q: How do I know which terms to simplify when multiplying two polynomial functions?

A: When multiplying two polynomial functions, we simplify the resulting expression by combining like terms. For example, if we have the expression $2x^3 - 14x + x^2 - 7$, we can simplify it by combining the like terms to get $2x^3 + x^2 - 14x - 7$.

Conclusion

In conclusion, multiplying two polynomial functions involves multiplying each term of one function by each term of the other function and then combining like terms. By following the steps outlined in this article, we can multiply two polynomial functions and write the resulting polynomial function in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

Example Solutions

Let's use the functions $f(x)=2x+1$ and $g(x)=x^2-7$ as examples to illustrate how to multiply two polynomial functions.

Example 1: Multiplying Two Polynomial Functions with Different Variables

$f(x) \cdot g(y) = (2x+1) \cdot (y^2-7)$

$= 2x \cdot y^2 - 2x \cdot 7 + 1 \cdot y^2 - 1 \cdot 7$

$= 2xy^2 - 14y + y^2 - 7$

Example 2: Multiplying Two Polynomial Functions with Different Degrees

$f(x) \cdot g(x) = (2x+1) \cdot (x^2-7)$

$= 2x \cdot x^2 - 2x \cdot 7 + 1 \cdot x^2 - 1 \cdot 7$

$= 2x^3 - 14x + x^2 - 7$

Example 3: Simplifying the Resulting Expression

$f(x) \cdot g(x) = 2x^3 - 14x + x^2 - 7$

$= 2x^3 + x^2 - 14x - 7$

Answer

The final answer is: $\boxed{2x^3 + x^2 - 14x - 7}$